What is meant by degrees of freedom of latent variables?
A good example is the degree of freedom in Student's distribution:

‌ The degrees of freedom refers to the number of independent observations in a set of data.

For example:

When estimating a mean score or a proportion from a single sample, the number of independent observations is equal to the sample size minus one.

e.g, if we have 100 observation $X_1, \ldots, X_{100}$ and we want to estimate their mean $\bar{X}$, as $\bar{X} = \frac{X_1 + \cdots + X_{100}}{100}$, if we know the mean, just we need to find the value of 99 variables from $X_1, \ldots, X_{100}$. Hence, here the degree of freedom is 99.
Your referenced paragraph is a general explanation in the paper as well. However, base on the above example, the degree of freedom in the paragraph depends on the likelihood function and number of observations that we have from spectrograms.
Now, as the DoF of latent variables is not high, using Gibbs sampling we will approximate some observations, and then using them we will compute the value of the latent variables.